Fröhlich polaron calculations in non-integer dimensional space as a model of confinement
Polaron is a quasiparticle describing an electron in interaction with phonons of a medium. A microscopic description of large polaron is given by the Fr¨ohlich Hamiltonian which does not admit exact solutions. For calculating the ground state energy and effective mass of polaron there are several approximation methods, some of which are valid only for large or small values of the electron-phonon coupling constant. In lowdimensional systems, where the polaron is confined by an external potential such as in the form of slab or wire geometries, the polaronic energy and effective mass are known to get enhanced. In this thesis we present an approach towards quantifying the degree of confinement on a large polaron provided by a parabolic potential. On that purpose, first, variation of polaronic ground state energy as a function of the parameters of the confinement potential for both slab- and wire-like geometries and using a methodology valid for all values of electron-phonon coupling constant is calculated. Then, applying a noninteger- dimensional-space algebra the polaron problem has been solved in an isotropic D-dimensional space using the same approch (D varies continuously from 3 to 2 for slab, and from 3 to 1 for wire geometry.) Finally, by matching the polaron ground state energy values obtained from the two calculations in large electron-phonon coupling constant limit, we identify the effective dimensionality D, of the polaron for a given set of confinement and material parameters.